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1996-29

Direct Galerkin-Approximation of the Channel Flow by Stokes Eigenfunctions

by Noske, A., Rummler, B..


Series: 1996-29, Preprints

MSC:
76F10 Shear flows
35Q30 ~Navier-Stokes equations
42C15 General harmonic expansions, frames
47A75 Eigenvalue problems
76H05 Transonic flows

Abstract:
We explore the plane parallel channel flows of incompressible Newtonian fluids
in an infinite layer of R 3 with a thickness of h. We presume non-slip conditions
of the fluid at the walls. We write the task as initial-boundary value problem
of the Navier-Stokes equations. Now we transform these equations to a non-
dimensionalized version. The limitation of the domain on an open bounded rect-
angular parallelepiped in R 3 furnished with periodical conditions for the sought
velocity field in the antecedent unbounded directions and the decomposition of
the velocity field in two parts affords equations for the determination of the
remaining velocity. The laminar velocity is fulfilling homogenous Dirichlet con-
ditions. The mainspring of flow is a constant pressure gradient. The application
of Galerkin method precipitates a corresponding system of ordinary differential
equations. The features of this system depend on one parameter Re (Reynolds
number based on the laminar velocity on the midline of the channel). For the
Galerkin method we utilize Stokes eigenfunctions and a fix period 2l = 2 \Theta 2; 69.
For our first considerations we use 356 eigenfunctions (to attain all eigenvalues
less then or equal to 4ß 2 ). For solving our problem in time we exert a Runge-
Kutta-Fehlberg method. We define a kinetic energy as an essential quantity for
the valuation of our solution. The results of computations lead to the conclu-
sions that some features of the laminar turbulent transition are describable by
our Galerkin system.

Keywords:
Navier-Stokes equations, plane channel flows, Galerkin method